The Minkowski question mark function ?(x) arises as a real distribution of rationals in the Farey tree. We examine the generating function of moments of ?(x). It appears that the generating function is a direct dyadic analogue of period functions for Maass wave forms and it is defined in the cut plane \ (1, ∞). The exponential generating function satisfies an integral equation with kernel being the Bessel function. The solution of this integral equation leads to the definition of dyadic eigenfunctions, arising from a certain Hilbert–Schmidt operator. Finally, we describe p-adic distribution of rationals in the Stern–Brocot tree. Surprisingly, the Eisenstein series G2(z) does manifest in both real and p-adic cases.

We complete the derived equivalence classification of all symmetric algebras of polynomial growth, by solving the subtle problem of distinguishing the standard and nonstandard nondomestic symmetric algebras of polynomial growth up to derived equivalence.

Existence theorems for the equation F(x)=\varphi (x) are proved when F is a function with “good” surjective properties and \varphi satisfies certain compactness conditions on countable subsets of the space. Also results for certain homotopic perturbations of the equation are obtained. The results lead to various fixed point theorems of Darbo type for F=id, but they are also applicable if F acts between different spaces. Also the inclusions F(x)\in \varphi (x) (resp. F(x)\subseteq \varphi (x)) for multivalued functions \varphi (resp. F and \varphi) are studied. There are some connections with the theory of 0-epi maps.

We investigate the constancy of the Milnor number of one parameter deformations of holomorphic germs of functions $f:(\C^n,0) \to (\C,0)$ with isolated singularity, in terms of some Newton polyhedra associated to such germs.

When the Jacobian ideals $J(\hspace*{1.8pt}f_t) = \left\langle {\partial f_t}/{\partial x_{1}} \ldots ,{\partial f_t}/{\partial x_{n}}\right\rangle $ of a deformation $f_t(x) = f(x)+ \sum_{s=1}^{\ell}\delta_s(t)g_s(x)$ are non-degenerate on some fixed Newton polyhedron $\Gamma_+$, we show that this family has constant Milnor number for small values of $t$, if and only if all germs $g_s$ have non-decreasing $\Gamma$-order with respect to $f$. As a consequence of these results we give a positive answer to Zariski's question for Milnor constant families satisfying a non-degeneracy condition on the Jacobian ideals.

Given a number field K, we find two simple separate necessary and sufficient conditions on a given algebraic number for it to be expressible as a quotient (respectively as a difference) of two algebraic numbers conjugate over K.

For a derivation δ of a commutative Noetherian ${\mathbb C}$-algebra A, a homeomorphism is established between the prime spectrum of the Ore extension A[z;δ] and the Poisson prime spectrum of the polynomial algebra A[z] endowed with the Poisson bracket such that {A,A}=0 and {z,a}=δ(a) for all a ∈ A.

Many of the techniques and notions used to study various important theorems in locally convex spaces are not effective for general linear topological spaces. In [4], a study is made of notionsin general linear topological spaces which can be used to replace barrelled, bornological, and quasi-barrelled spaces. The present paper contains a parallel study in the context of semiconvex spaces.

A. W. Chatters and D. A. Jordan defined in [0] a unique factorization ring to be a prime ring in which every height one prime ideal is principal. In this note we will prove that the trace ring of m generic n × n-matrices satisfies this condition.

Throughout this note, k will be a field of characteristic zero. Consider the polynomial ring S = k[;1≤i, j≤n, 1≤l≤m] and the n × n matrices Xl = in Mn(S). The k-subalgebra of Mn(S) generated by {Xl; 1≤l≤m} is called the ring of m generic n × n matrices Gm,n. Adjoining to it the traces of all its elements we obtain the trace ring of m n × n generic matrices, cfr. e.g. [1].

Bellman [1], [2, p. 116] proved that, if all solutions of the equation

are in L2, ∞) and b(t) is bounded, then all solutions of

are also in L2(a, ∞). The purpose of this paper is to present conditions on the function f that guarantee that all solutions of

be in the class L2(a, ∞) whenever all solutions of the equation

have this property. It is assumed that r(t) >0, r and qare continuous on a half line (a, ∞) and f is continuous. Actually the continuity assumptions may be weakened to local integrability and L2 (a, ∞) may be replaced by Lp(a, ∞) for any p > 1.

Let X be an F-space (complete metric linear space) and suppose g:[0, 1] → X is a continuous map. Suppose that g has zero derivative on [0, 1], i.e.

for 0≤t≤1 (we take the left and right derivatives at the end points). Then, if X is locally convex or even if it merely possesses a separating family of continuous linear functionals, we can conclude that g is constant by using the Mean Value Theorem. If however X* = {0} then it may happen that g is not constant; for example, let X = Lp(0, 1) (0≤p≤1) and g(t) = l[0,t] (0≤t≤1) (the characteristic function of [0, t]). This example is due to Rolewicz [6], [7; p. 116].

In this note, we give a homology-free proof that the non-abelian tensor product of two finite groups is finite. In addition, we provide an explicit proof that the non-abelian tensor product of two finite p-groups is a finite p-group.

We introduce and study the type I-, II-, and III-Λ-complete continuity property of Banach spaces, where Λ is a subset of the dual group of a compact metrizable abelian group G.

A ring $R$ with identity is called strongly clean if every element of $R$ is the sum of an idempotent and a unit that commute. For a commutative local ring $R$, $n=3,4$, and $m, k, s \in {\mathbb N}$ it is proved that ${\mathbb M}_n(R)$ is strongly clean if and only if ${\mathbb M}_n(R[[x]])$ is strongly clean if and only if ${\mathbb M}_n(R[[x_1, x_2, \ldots, x_m]])$ is strongly clean if and only if $ {\mathbb M}_n(\frac{R[x]}{(x^{k})})$ is strongly clean if and only if $ {\mathbb M}_n(\dfrac{R[x_{1}, x_{2}, \ldots , x_{s}]}{(x^{n_1}_{1}, x^{n_{2}}_{2}, \ldots , x^{n_{s}}_{s})}) $ is strongly clean if and only if ${\mathbb M}_n(R \propto R)$ is strongly clean where $ R\propto R=\{\scriptsize(\begin{array}{@{}c@{\quad}c@{}} a& b \\ 0& a \end{array}): a, b \in R \}$ is the trivial extension of $R$. This extends a result of J. Chen, X. Yang and Y. Zhou [$\mathbf{5}$] from $n=2$ to 3 and 4.

We study the semilinear differential equation u″ + F(t,u,u′)=0 on a half-line. Under different growth conditions on the function F, equations with globally defined solutions asymptotic to lines are characterized. Both fixed initial data and fixed asymptote are considered.

We present several conjectures which would describe the Nielsen equivalence classes of generating pairs for the groups SL(2,q) and PSL(2,q). The Higman invariant, which is the union of the conjugacy classes of the commutator of a generating pair and its inverse, and the trace of the commutator play key roles. Combining known results with additional work, we clarify the relationships between the conjectures, and obtain various partial results concerning them. Motivated by the work of Macbeath (A. M. Macbeath, Generators of the linear fractional groups, in Number theory (Proc. Sympos. Pure Math., vol. XII, Houston, TX, 1967) (American Mathematical Society, Providence, RI, 1969), 14–32), we use another invariant defined using traces to develop algorithms that enable us to verify the conjectures computationally for all q up to 101, and to prove the conjectures for a highly restricted but possibly infinite set of q.

Considered here is the pullback attractor of the process associated with the first initial boundary value problem for the non-autonomous semilinear degenerate parabolic equation

\begin{linenomath} u_t-\text{div}(\sigma(x)\nabla u)+f(u)=g(x,t) \end{linenomath}

$D_0^2(\Omega,\sigma)$

Let G be a finite additive abelian group, and suppose that A and B are subsets of G. We say that G = A⊕B if every element g ∈ G can be uniquely written in the form g = a + b, where a ∈ A, b ∈ B. The study of such decompositions (usually called factorizations in the literature) was initiated by G. Hájos [3] in connection with his solution to a problem of Minkowski in the geometry of numbers.

This paper is a continuation of [3] in which some inequalities for the Schatten p-norm were considered. The purpose of the present paper is to improve some inequalities in [3] as well as to give more inequalities in the same spirit.

Let H be a separable, infinite dimensional complex Hilbert space, and let B(H) denote the algebra of all bounded linear operators acting on H. Let K(H) denote the closed two-sided ideal of compact operators on H. For any compact operator A, let |A| = (A*A)½ and s1(A), s2(A),… be the eigenvalues of |A| in decreasing order and repeated according to multiplicity. A compact operator A is said to be in the Schatten p-class Cp(1 ≤ p < ∞), if Σ s1(A)p < ∞. The Schatten p-norm of A is defined by ∥A∥p = (Σ si(A)p)1/p. This norm makes Cp into a Banach space. Hence C1 is the trace class and C2 is the Hilbert-Schmidt class. It is reasonable to let C∞ denote the ideal of compact operators K(H), and ∥.∥∞ stand for the usual operator norm.

The integral equation of the title is

It was studied in [4], though h(x) was written as x-1g(x-1) there, and using a method involving orthogonal Watson transformations, it was shown there that if h ∈ L2(0, ∞), then the equation has a solution f ∈ L2(0, ∞), and that / is given by

In this paper, using the techniques of [3], we shall show that the equation can be solved for ℎ in the space ℒμ, p of [3] for 1 ≤ p < ∘, μ > 0, and that for these spaces, which include L2(0, ∘), f is given by the simpler formula

We shall further show that these results can be extended to the spaces ℒw, μ, p of [3]. This forms the content of our theorem below.

Let ℱ be a smooth foliation of codimension p on a smooth manifold Mm. We can define ℱ by an atlas of coordinate charts (U, (x, y)), called leaf charts, where (x, y): U → Rm−p × Rp are coordinate functions for which the leaves of ℱ are given by y1 constant,…,yp constant, in U. Clearly, on the overlap of two such leaf charts (U, (x, y)) and (U′, (x′, y′)) we have a coordinate transformation of the form

If y′ is always affine in y, i.e.

where and Bi are constants, we shall say that ℱ is a transversally affine foliation. This notion is, in a sense, dual to that of affine foliation, see [2], in which x′ is affine in x and each leaf has an induced flat affine structure.